When Teachers Break Rules

Submitted by DavidSpangler on Fri, 01/20/2012 - 11:08

When Teachers Break Rules

by David B. Spangler

 

This "humorous" story makes an important point about teachers who tell students “rules” that are ultimately “broken” by other teachers in later grades. Here are some examples:

When teaching subtraction with renaming (as in 85 – 37), a third-grade teacher may say, “You cannot subtract 7 from 5 because 7 is greater than 5. You cannot take a larger number from a smaller number.” But when the student reaches sixth grade, he or she learns that with integers you can take 7 from 5. The third-grade teacher should have said something akin to: “We don’t have enough ones to take 7 ones from 5 ones, so we rename 8 tens and 5 ones as 7 tens and 15 ones.”

When attempting to simplify an expression such as 3a + 4b, a teacher may explain: “You cannot combine 3a and 4b because you cannot add apples and oranges.” However, later students learn that they can multiply 3a and 4b (obtaining 12ab). So some students may ask, “Why is it that we can multiply apples and oranges, but we cannot add them?” At this point one must wonder: “Do the apples and oranges still have ‘a-peel’? Did this metaphor provide a ‘fruitful’ experience?” Note that rather than discussing apples and oranges, it should be explained that the distributive property enables us to add like terms: 3a + 4a = (3 + 4)a = 7a. However, we cannot use the distributive property with 3a + 4b because a and b are unlike terms.

An Algebra I teacher may say, “We cannot factor x2 + 1.” But in Algebra II, the student learns that you can factor x2 + 1 by using imaginary numbers: (x + i ) (x – i ). The Algebra I teacher should have said, “You cannot factor x2 + 1 over the real numbers.”

Additional "rules" that are later broken include: "Multiplication always makes larger" (broken when students learn that 3/4 • 1/2 = 3/8; the product is less than either of the two factors) and "Division always makes smaller" (broken when students learn that 2 ÷ 1/8 = 16). 

The story picks up with a dialogue between a teacher and a sixth-grade student who has already been exposed to a few of these “broken rules.”

Teacher: Division by 0 is impossible. You cannot divide by 0.

Student: Will we be able to divide by 0 when we’re in seventh grade?

Comments

beatydon's picture

beatydon

Wed, 01/25/2012 - 21:16

Even multiplication by one does not result in a larger value. And, of course division by one does not produce a smaller value.

Teaching at the junior high level, I was the teacher that had to break (or expose the myth of) the mentioned rules. To avoid doing the same thing to someone down the line, talking about using two symbols for multipication, I would say: 

With the numbers that we are using, the symbols x and • both mean the same thing (multiply) and produce the same result.  Later you may study something called vectors, basically numbers that include direction, for which the two symbols can give different results. One is called a cross product and the other a dot product. But for now, we don't have to worry about it. I even know of a third kind of product called a convolution product (symbol, *)where it's possible to multiply a square times a triangle. You might have a lot of fun with that someday. So for us today, the symbols x, • and * all mean to multiply.